Relative frequency probability

The second type of probability, relative frequency probability, is calculated by repeating an experiment a large number of times (say ) and counting the number of times out of that the outcome of interest (say m) occurred. The probability of the event is then calculated as ... 

The probability of finding a particle with a molecular speed somewhere between 0 and oo is 1.0 because negative molecular speeds are impossible hence, the relative frequency of speeds in excess of v is 1.0 — Jq G v)dv. 

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

When X represents a continuous variable quantity, it is sometimes convenient to take the total or relative frequency of occurrences within a given range of x values. These frequencies can then be plotted against the midvalues of x to form a histogram. In this case, the ordinate should be the frequency per unit of width x. This makes the area under any bar proportional to the probability that the value of x will he in the given range. If the relative frequency is plotted as ordinate, the sum of the areas under the bars is unity. 

The opinions of the experts, however obtained, provide a basis for plotting a frequency or probability distribution curve. If the relative Frequency is plotted as ordinate, the total area under the cui ve is unity. The area under the cui ve between two values of the quantity is the probability that a randomly selected value will fall in the range between the two values of the quantity. These probabilities are mere estimates, and their reliability depends on the skill of the forecasters. 

The area under the cui ve of f z) is unity if the abscissa extends from minus infinity to plus infinity. The area under the cui ve between Z and Zo is the probability that a randomly selected value of x will lie in the range Z and r2, since this is the relative frequency with which that range of values would be represented in an infinite number of trials. 

Suppose that Uie valve slicks twice as often in Uie open position as it does in Uie closed position. Under Uie Uieorelical relative frequency interpretaUon, Uie probability assigned to element 0 in S would be j, twice the probability... 

For example, consider Uie random experiment of drawing two cards in succession from a deck of 52 cards. Suppose Uie cards are drawn wiUiout replacement (i.e., Uie first card drawn is not replaced before the second is drawn). Let A denote Uie event Uiat Uie first card is an ace and B Uie event Uiat Uie second card is an ace. The sample space S can be described as a set of 52 times 51 pairs of cards. Assuming Uiat each of these (52)(51) pairs lias Uie same Uieoretical relative frequency, assign probability 1/(52)(51) to each pair. The number of pairs featuring an ace as Uie first and second card is (4)(3). Therefore,... 

A statistical method for plotting the relative frequency (dN/N) of a probable error in a single measured value X versus the deviation (z) from fi, the mean of the data, in units of standard deviation (o-), such that z = (x -fji)/a. The standard error curve (shown below) does not depend on either the magnitude of the mean or the standard deviation of the data set. The maximum of the normal error curve is poised at zero, indicating that the mean is the most frequently observed value. 

In practice, various complications may be encountered for which the simphstic description above will not be adequate. First, still within the realm of ID variabihty modeling, the measurements may be in some sense partially missing, e.g., censored or available only as summary statistics. In addition, methods may be applicable for specifying distributions based on professional judgment, particularly where the probabihties of interest do not represent relative frequencies, or the probabilities of interest do represent relative frequencies, but there are inadequate data to justify particular distributions. 

The topic of eliciting probability distributions that are based purely on judgment (professional or otherwise) is discussed in texts on risk assessment (e.g., Moore 1983 Vose 2000) and decision theory or Bayesian methodology (e.g., Berger 1985). Elicitation methods may be considered with ID models in case no data are available for htting a model. In the 2D situation, elicitation may be used for the parameter uncertainty distribntions. In that situation, it may happen that no kind of relative fre-qnency data wonld be relevant, simply because the distributions represent subjective uncertainty and not relative frequency. 

Fig. 7.4. The frequency of observed O-H... O geometries projected onto the plane of the three atoms. The O-H bonds lie on the horizontal axis. The contours show the probability of finding the acceptor ion, each contour representing a doubling of the probability relative to the contour below. The broken line shows the closest approach observed for acceptor O atoms (adapted from Brown 1976a).

Probability is a number indicating either the relative frequency of an event (e.g., the chance that a person randomly selected from a population will be age 30 years) or the chance that something is true (e.g., the subjective probability that humans are more sensitive to a particular substance than mice and rats). 

In practice, p., X , and p. have to be estimated. Classical quadratic discriminant analysis (CQDA) uses the group s mean and empirical covariance matrix to estimate Pj and X. The membership probabilities are usually estimated by the relative frequencies of the observations in each group, hence pc = n. In, where nj is the number of observations in group j. 

The so-called probability assumptions. In the motion of molecules, which is too complicated to be observed, certain regularities are described in terms of statements about the relative frequency of various configurations and motions of the molecules.3 (Cf. Sections 3-5.)... 

Subsequently, however, this probability is interpreted to mean either the ratio of time intervals or the relative frequencies in other, very different statistical ensembles. In this way formulation (I ) leads to statement (III) and also to all other further statements which, together, make up the modified formulation of the H-theorem. 

In both the conjugate and nonconjugate cases just described, the most natural estimator of the posterior probability of a subset indicator vector 8 is the observed relative frequency of 8 among the K sampled subsets S = 61, 82,..., 6 that is,... 

Here, the indicator function /(.) is 1 whenever its argument is true, and zero otherwise. A number of problems arise with the relative frequency estimate (15). First, it is prone to variability in the MCMC sample. Second, any model that is not in S has an estimated posterior probability of zero. Third, if the starting value 6° has very low posterior probability, it may take the Markov chain a large number of steps to move to 6 values that have high posterior probability. These initial burn-in values of 8 would have larger estimates of the posterior probability (15) than their actual posterior probability that is, the estimate p(8 Y) will be biased because of the burn-in. For example, with the simulated data described in Section 4.2, the first 100 draws of 6 have almost zero posterior probability. In a run of K = 1000... 

In conjugate settings such as (4) or (6), a better estimate of the posterior probability p(8 Y) is available. Instead of the relative frequency estimate (15), the analytic expression for the posterior probability (14) is used that is,... 

In some situations, such as when estimates of the posterior probability of a specific subset 6 or of groups of subsets are required, a second method of estimating C can be used. Notice that the estimate C in (17) will be biased upwards, because C = C in (17) only if U = V, the set of all possible values of 6. If U c V, then C < C. A better estimate of C can be obtained by a capture-recapture approach, as discussed by George and McCulloch (1997). Let the initial capture set A be a collection of 6 values identified before a run of the MCMC search that is, each element in the set A. is a particular subset. The recapture estimate of the probability of A is the relative frequency given by (15). The analytic expression (16) for the posterior probability of A is also available, and contains the unknown C. Let g(A) = J2seA8( ) so that p(A Y) = Cg(A). Then, by equating the two estimates, we have... 

Having seen that the observed frequency distribution of the sampler is useful in the estimation of the normalizing constant C, it is interesting to note that the frequencies are otherwise unused. Analytic expressions for the posterior probability are superior because they eliminate MCMC variability inherent in relative frequency estimates of posterior probabilities. In such a context, the main goal of the sampler is to visit as many different high probability models as possible. Visits to a model after the first add no value, because the model has already been identified for analytic evaluation of p(6 Y). 

Figure 7 also illustrates the burn-in problem with relative frequency estimates of posterior probability which was discussed in Section 3.2. The first 100 iterations of the algorithm visit improbable subsets, so a relative frequency estimate of posterior probability (15) will place too much probability on these 100 subsets. 

Probability of occurrence Estimate of relative frequency, which can be discrete or continuous. 

Probability of occurrence Estimate for the relative frequency of a discrete or continuous loss of function Probability... 

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